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Having the ability to solve equations is useful. Randomly write down an equation. That particular equation isn't likely to have practical applications. Is it worthwhile to explore whether or not it can be solved by algebraic methods?

Why is it that first, second, third, and fourth degree polynomials can be solved by algebraic methods but not arbitrary polynomials of higher degree? Is this worth studying only if it has practical applications? It does have applications today but it didn't for the first thousand years this problem was tackled. The requirement that something be 'useful' before it is worthy to be studied is too great a burden. It's a detriment to intellectualism.